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MECH-210

STATICS

Chapter 2

Concurrent Force Systems

COURSE INFORMATION

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Fundamental Concepts

Force action of one body on another. Where do forces comefrom?

!Direct contact (surface)" Concentrated force" Distributed force

!Non-contact (body)" Gravitational" Electrical" Magnetic

Effect of force on a body!External effect: change of motion! Internal effect: deformation

Force system a number of forces treated as a group!No external effect -> Forces in balance -> body in equilibrium!change of motion must occur if a force system is unbalanced (has a

resultant)

Statics Forces in balance -> Body in equilibrium -> No external effect

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Fundamental Concepts

Concurrent forces: a set of forces which all pass through thesame point

Coplanar forces: if forces lie in the same plane Parallel forces: the lines of action are parallel Collinear forces: forces with the common line of action

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Resultant of Two Concurrent Forces

Force: action of one body on another;characterized by its

!magnitude!direction (line of action&sense), and!point of application

Force is a vectorquantity. The resultant is equivalent to the diagonal of a

parallelogram which contains the two forces in

adjacent legs.

Transmissibility: for rigid body, only theexternal effects of any force Force are of

interest. So the force can be applied at anypoint along its line of action.

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Resultant of Two Concurrent Forces

Parallelogram rule for vector addition Triangle rule for vector addition

B

B

C

C

QPR

BPQQPR!!!

+=

!+= cos2222

Law of cosines,

Law of sines,

B

R

C

Q

A

P

sinsinsin==

Example problems 2-1, 2-2.

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Resultant of 3 or More Concurrent Forces

Resultant of three or more concurrent forcesthrough repeated application of the triangle

rule

The polygon rule for the addition of three ormore vectors.

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Resolution of a Force into Components

Reverse process of finding the resultant A single force can be replaced by a system of 2

or more forces (components)

R

!

A

!

B

!

D

!

C!

DCR!!!

+=

BAR!!!

+=

Resolution process > non unique sets ofvector components

Use parallelogram & triangle laws to resolvea force into components along any directions

(coordinates).

Example problems 2-4, 2-5.

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Rectangular Components of a Force

R

!

A

!

B

!

D

!

C!

A force can be resolved into componentsalong any oblique lines, however, rectangular

coordinates are used the most

F!

xF

!

yF!

x

y

!

jFiFFFF yxyx!!!!"

+=+=

x

y

yx

y

x

FF

FFF

FF

FF

1

22

tan

sin

cos

!=

+=

=

=

"

"

"

2-D

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Rectangular Components of a Force

Fzyx

zyxzyx

eFkFjFiF

kFjFiFFFFF

!

!!!

!!!!!!"

=++=

++=++=

!!! coscoscos

222

cos

cos

cos

zyx

zz

yy

xx

FFFF

FF

FFFF

++=

=

=

=

!

!

!

kjie zyxF!!!

!

!!! coscoscos ++=

3-D

F!

xF

!

yF!

zF

!

Fe!

Use Dot (Scalar) product to the determineforce components

nnn

xzyxx

FeFF

FikFjFiFiFF

!

!

cos

cos)(

="=

="++="=

!

!

!!!!!!

Example problems 2-7, 2-8.

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Resultant by Rectangular Components

SQPR!!!!

++=

Wish to find the resultant of 3 or moreconcurrent forces,

( ) ( )jSQPiSQPjSiSjQiQjPiPjRiR

yyyxxxyxyxyxyx !!

!!!!!!!!

+++++=

+++++=+

Resolve each force into rectangular components

!=++=

xxxxx

FSQPR

The scalar components of the resultant are equalto the sum of the corresponding scalar

components of the given forces.

!=++=

y

yyyyF

SQPR

x

yyx

R

RRRR

122tan

!

=+= "

To find the resultant magnitude and direction,

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Sample Problem

Four forces act on boltAas shown.

Determine the resultant of the force

on the bolt.

SOLUTION:

Resolve each force into rectangularcomponents.

Calculate the magnitude and directionof the resultant.

Determine the components of theresultant by adding the correspondingforce components.

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Sample Problem

SOLUTION:

Resolve each force into rectangular components.

9.256.96100

0.1100110

2.754.2780

0.759.129150

4

3

2

1

!+

!

+!

++

!!

F

F

F

F

compycompxmagforce

!

!

!

!

223.141.199 +=R N6.199=R

Calculate the magnitude and direction.

N1.199

N3.14tan =! = 1.4!

Determine the components of the resultant byadding the corresponding force components.

1.199+=xR 3.14+=yR